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Trainer pack
Module 7: Developing the
personal maths skills of teachers
and assessors
Shape and space master class
Learning and Skills Improvement Service
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Course
information
Length of session: approx. 5hr 30 min, not including lunch or
breaks.
Trainers can customise the session to suit the audiences and
settings.
The session is written to be delivered as one full day, but could
equally be delivered as 2 half-days.
Audience
Job roles: Practitioners who are teaching or supporting adult
numeracy on functional skills, embedded and discrete
programmes, and who wish to improve their understanding
and application of number at Level 3.
Sector / setting: any part of the learning and skills sector.
Links to other
modules
This session forms part of a blended learning programme,
aimed at preparing participants for the Level 3 Award in
Mathematics for Numeracy Teaching.
Aims
To revise / widen participants’ personal mathematical skills relating to geometry and
trigonometry
Outcomes
By the end of the session participants will have:

reflected upon personal maths skills and identified areas for development;

revised, consolidated and applied geometric properties and relationships; and

identified and applied appropriate concepts of shape and space in order to
represent and solve problems.
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Session overview
Activity
1
Content
Progress review, aims
and outcomes
4
Participants to feed back on progress with course and
adopt solution-focused approach to challenges. Group
discussion of incidence of shape and space in private,
academic and vocational situations.
Diagnostic
Participants engage in a variety of activities exploring
assessment –
concepts of shape and space. Through investigation and
carousel of activities
discussion they build on their awareness of personal
development needs.
Geometric figures and Direct teaching, revision and consolidation of underpinning
their properties
knowledge.
Length and area
Participants use geometric properties to solve problems.
5
Construction activity
6
Area and volume of
3D shapes
Pythagoras and
simple trigonometry
2
3
7
8
9
10
Pythagoras and
simple trigonometry
presentation
Consolidation
questions
11
Using concepts of
shape and space to
solve problems
Developmental tasks
12
Plenary and close
(Optional). Participants engage in a variety of geometric
construction activities.
Direct teaching, revision and consolidation of underpinning
knowledge.
Assessment / consolidation activity in which participants
design posters to display their knowledge of underpinning
skills.
Direct teaching, revision and consolidation of underpinning
knowledge.
Participants practise the selection and application of the
appropriate formulae to calculate sides and angles in
simple triangles.
Participants work in small groups on a variety of
differentiated problem solving tasks requiring the
application of concepts of shape and space.
Whole group activity to examine the three developmental
tasks associated with this master class. Ideas are shared
and approaches discussed. Support materials are
identified.
Revisit assessment criteria for Level 3 course.
Revisit aims and outcomes.
Check on individual progress.
Trainers
Trainer experience or
qualifications required
Fully qualified numeracy teacher (Cert Ed / PGCE or
equivalent plus numeracy subject specialist
qualification); minimum Level 3 mathematics
qualification.
Reference material for
trainers
Course materials and trainer notes.
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Participants
Prior knowledge and
qualifications
Participants should have a Level 2 maths qualification
as a minimum requirement, and some familiarity with
basic algebra and trigonometry.
Pre-course activity for
participants
All participants should have completed the initial
assessment and self-assessment checklist, and have
attended the course induction session. Ideally they
should bring along their ILP, as it will be reviewed
during the session. Remind participants that drawing
instruments will be needed for this session.
Resources
Resources for
reference during the
session
Before the session the
trainer needs to:
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As provided or any suitable alternatives chosen by the
trainer.
Prepare all resource materials. These are included
below. Note that the PowerPoint is a separate file. A
set of PowerPoint slide notes is included in the
Participant pack. The PowerPoint is taken from the
Resource pack for Level 3 preparation programmes
for entry to numeracy teacher training:
http://archive.excellencegateway.org.uk/media/Skills%
20for%20Life%20Starter%20Toolkits/3.6_Level_3_Ad
vanced_Numeracy_Preparation_Resource_Pack.pdf

Prepare all handouts in advance – these can be
distributed to participants as they arrive.

Note that correct answers are provided for
resources where this is appropriate. Some
resources are open tasks and do not have
answers.

Prepare resources (R 1 – R 4) for carousel activity
in TN 2.

Prepare (cut up) resource cards for R 5, R 7.

TN 5 is an optional activity about drawing shapes;
drawing instruments will be required.

TN 6 is an optional revision of formulae. The trainer
will need to put this together, based on participants’
needs – note that this type of material is readily
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available from a number of sources.

TN 7 – paper and pens will be needed for
participants to create posters.

TN 8: set up the PowerPoint presentation.

TN 9: two of the resources (R 8 and R 9) include
the correct responses – you will need to separate
these and hand out the correct response at an
appropriate time.

TN 10: the resources used here (R 10, R 11, R 12)
all include the correct responses – you will need to
separate these and hand out the correct response
at an appropriate time.

TN 11: this introduces the three developmental
tasks associated with this master class. Either you
will need to print off copies of these, or (preferably)
show participants where they are online. For the
second option you will need access to the internet.

It might be an idea to take along some drawing
instruments in case participants forget to bring
them.
Session plan
Aims
To revise / widen participants’ personal mathematical skills relating to geometry and
trigonometry
Outcomes
By the end of the session participants will have:

reflected upon personal maths skills and identified areas for development;

revised, consolidated and applied geometric properties and relationships; and

identified and applied appropriate concepts of shape and space in order to
represent and solve problems.
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Suggested timings are for guidance purposes only. Trainers should adapt the
content to reflect the results of initial assessment and to meet the needs and
experience levels of the participants.
TN – Teacher notes HO – handout
Time
15m
R – resource
Content
PPT – PowerPoint
No.
Resources
Style
Title
HO 1
Handout
Reflection
sheet
R1-R4
Resourc
es
Carousel of
activities
TN 1. Welcome, progress review, aims
and outcomes
Purpose of this activity: to introduce the
aims and intended outcomes of the
session and begin to explore aspects of
shape and space.
Each person to give account of progress
since the Induction workshop and previous
master classes (for Number and Algebra).
Review of developmental tasks
undertaken since last master class.
Activity 1
On their table, participants discuss how
they encounter space and shape in their
own private, professional or academic
lives. If they find this difficult then broaden
the discussion to include vicarious
knowledge and experience of maths
usage, e.g. by learners.
(Total
15m)
Introduce the reflection sheet for this
session – HO 1. Participants will also need
their own ILPs, in order to amend this if
necessary.
30m
TN 2. Assessment activity
Purpose of this activity: to give
participants an opportunity to explore
shape and space in a variety of ways, also
allowing the trainer to observe participants’
level of skills and confidence.
Activity 2
Carousel of activities and investigations.
These should now be differentiated
according to original initial assessment,
ILPs and trainer’s knowledge of individual
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Time
Content
No.
Resources
Style
Title
participants.
Trainers may develop their own or use /
adapt resources R 1 – R 4.
Ensure there are sufficient activities for the
group size. Encourage participants to work
together to problem solve and discuss
their approaches.
 R 1: Pentominoes
(Total
45m)

R 2: Classifying shapes (7 pages)

R 3: Dissecting a square

R 4: Transformations
R 2: note that this is an open task so no
solutions are given. With Card set A –
Shapes, ask participants to sort the
shapes into two groups using criteria of
their own choice. Next, ask them to sort
each group into two, using further criteria.
Give out blank cards and ask them to write
a description of each of their four groups
and also to draw another shape to add to
each group.
Extension: Give each pair one of the
Sheets 1 to 5. Ask them to place shapes
into appropriate cells. Sometimes, several
shapes may go in a cell. If participants feel
that a cell is impossible to fill, they should
explain why this is so.
Parts of R 4 are also an open task.
Answers are provided for R 1 and R 3.
Give participants time to reflect on what
they need to do to improve their progress
and make notes on ILP and reflection
sheet – HO 1.
45m
TN 3. Geometric figures and their
properties
Purpose of this activity: to give
participants an opportunity to explore in
more detail the properties of shapes and
concepts.
Activity: ‘Just a minute’
Depending on the size of the group,
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R5
Resourc
e card
Geometric
figures and
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Time
Content
No.
participants may work as pairs or teams.
They take it in turns to select a card (made
from R 5) which is shown to all. Everyone
has 1 minute to prepare information
relating to the subject on the card. The
‘selectors’ then have 1 minute to impart all
they can about the subject. When the time
is up others may ‘challenge and correct’ or
add. Points may be awarded as the trainer
sees fit.
Resources
Style
Title
activity
concepts
R6
Resourc
e cards
Evaluating
statements
about area
and
perimeter
R7
Resourc
e cards,
cut up
Hints for
solutions
about area
and
perimeter
statements
This section will be dependent on the
underpinning skills of participants.
The trainer should then ‘top up’ on each
subject as appropriate by direct teaching
of aspects not covered.
(Total
1h
30m)
45m
Give participants time to reflect on what
they need to do to improve their progress
and make notes on ILP and reflection
sheet.
TN 4. Evaluating length and area using geometric properties to solve
problems
Purpose of this activity: to give
participants an opportunity to evaluate
statements about area and perimeter.
Ask participants to work in pairs. Give R 6
to each pair and ask them to choose a
statement to work on.
Differentiate by allocating statements if
necessary.
Participants should try to decide whether
their chosen statement is always,
sometimes or never true. Remind them of
the need to test specific examples in an
organised way. This testing may
necessitate accurate drawing in order to
prove the point.
If participants get stuck, offer the
appropriate card from R 7.
Repeat above with another statement.
Feedback
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Time
Content
No.
Resources
Style
Title
Invite pairs to describe one problem that
they think they have solved successfully
and the reasoning that they employed.
Then ask other pairs who have solved the
same problem to show their reasoning.
Ask the remaining participants to say
which reasoning they found most clear
and convincing and why.
(Total
2h
15m)
15m
Go through each one to ensure that
misunderstandings have been clarified
and underpinning skills widened. Identify
the problem solving steps used.
Allow time for reflection.
TN 5. Drawing and construction
(optional extension activity)
Purpose of this activity: to extend
thinking about properties of shapes to
practical drawing of shapes.
(Total
2h
30m)
30m
As an addition to the activity above it may
be helpful to explore participants’
knowledge and skill of drawing geometric
shapes – e.g. triangle given 3 sides,
concentric circles, circle within a square,
bisecting a line and so on.
Appropri
ate
drawing
instrume
nts
TN 6. Area and volume of 3D shapes
Purpose of this activity: optional revision
of formulae.
Direct teaching
At the trainer’s discretion, revise formulae
for area and volume of circles, spheres,
cylinders, etc, and display for all to see.
Formulae sheets are readily available and
may be provided as back up.
Optional
resource:
formulae
sheet
Revise changing the subject of a formula,
building on work done on equations in the
algebra master class and self-study
materials.
(Total
3h)
20m
Allow time for reflection and updating of
reflection sheet and ILP.
TN 7. Pythagoras and simple
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Time
Content
No.
Resources
Style
Title
HO 2
Handout
Trigonometry
information
sheet
Slides
Pythagoras
and
trigonometry
trigonometry
Purpose of this activity: an assessment
and consolidation activity.
(Total
3h
20m)
In small groups, each group prepares a
poster to display their underpinning
knowledge and understanding. Ensure
groups are fairly evenly split with half
doing one poster and half the other:
Poster 1 – Pythagoras
Poster 2 – Simple trigonometry
Paper
and pens
for
posters
Display posters and take feedback as
required. Adopt a ‘challenge and correct’
and addition approach to feedback. Take
feedback from one group at a time and
ask the remaining group if they want to
challenge or correct anything that has
been said. Then ask the other group if
they have anything they can add. Then
reverse the process. This approach allows
for full assessment of the entire group and
maximises the opportunity to uncover any
misconceptions or misunderstandings.
Give HO 2 for reference.
Allow time for reflection and updating of
reflection sheet and ILP.
20m
TN 8. Pythagoras and trigonometry
presentation
Purpose of the activity: presentation and
teaching of Pythagoras and trigonometry
(Total
3h
40m)
Use PPT 1-13 to explain Pythagoras.
Encourage all to contribute and draw out
points related to:
 Pythagorean triples
 Identification of sides
 Properties of all similar shapes –
revise ratio of side length, area and
volume (refer to workshop session on
proportional reasoning)
 Significance of similar triangles for trig
ratios
 Trig ratios and mnemonics for
remembering them – ask participants
(SOHCAHTOA)
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PPT 113
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Time
Content
No.



Resources
Style
Title
Use of scientific calculator for looking
up sin, cos, tan
Use of inverse key
Manipulation of formulae
Record for all to see.
Allow time for reflection and updating of
reflection sheet and ILP.
20m
(Total
4h m)
TN 9. Consolidation questions
Purpose of the activity: consolidation of
learning.
Differentiated groups work through
examples to reinforce calculations of sides
and angles: R 8 Pythagoras, R 9
Trigonometry and R10 The sine rule.
Note that R 8 and R 9 include the
answers.
45m
R8
Question
sheets
Pythagoras
questions +
answers
R9
Trigonometry
questions +
answers
R 10
The sine rule
TN 10. Using concepts of shape and
space to solve problems
Purpose of the activity: focus on problem
solving and relevance to functional skills.
(Total
4h
45m)
R 11
Work in small groups to solve problems.
Using geometric principles to solve
problems – R 11:
 Hen house
 Concrete posts
 Running track
Using trigonometry and Pythagoras to
solve problems – R 12
 Ice rink
 Paragliding
and R 13: When the boat comes in
Problem
sheets
Using
geometry to
solve
problems
R 12
Using
trigonometry
to solve
problems
R 13
When the
boat comes
in
Support with use of calculators, changing
units, etc.
Review problems as whole group and
draw out the problem solving techniques
used. Discuss different approaches.
Discuss implications for functional maths,
i.e. the focus on problem solving.
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Time
Content
No.
30m
TN 11. Developmental tasks
(Total
5h
15m)
Purpose of the activity: to introduce the
developmental tasks associated with this
master class.
Whole group activity to examine the three
developmental tasks associated with this
master class. Ideas are shared and
approaches discussed.
Resources
Style
Title
Develop
mental
tasks for
Shape
and
space
master
class
Support materials, for the tasks, or for
personal study, are identified.
10m
TN 12. Plenary and close
(Total
5h
00m)
Revisit aims and outcomes.
Review self-assessment checklist (from IA
session) and/or ILPs.
Complete final part of reflection sheet (HO
1).
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Handouts and resources for shape and space master class
Note that the resources need to be prepared by the trainer and the handouts are
grouped together, for ease of printing.
Note also that many of the resources have answers / solutions (see separate
list below) which you may want to hand out separately.
Handouts
HO 1: Space and shape reflection sheet
HO 2: Trigonometry information sheet
Resources
R 1: Pentominoes
R 2: Classifying shapes
R 3: Dissecting a square
R 4: Transformations and extension
R 5: Geometric figures and concepts
R 6: Evaluating statements about area and perimeter
R 7: Hints for solutions about area and perimeter statements
R 8: Pythagoras’ theorem – consolidation questions
R 9: Trigonometry calculations – practice questions
R 10: The sine rule
R 11: Using geometry to solve problems
R 12: Using trigonometry to solve problems
R 13: When the boat comes in
Resources - ANSWERS
R 1: Pentominoes ANSWERS
R 3: Dissecting a square ANSWERS
R 4: Transformations
ANSWERS
R 6 / R 7: Evaluating statements about area and perimeter ANSWERS
R 8: Pythagoras’ theorem – consolidation questions ANSWERS
R 9: Trigonometry calculations ANSWERS
R 10: The sine rule ANSWERS
R 11: Using geometry to solve problems ANSWERS
R 12: Using trigonometry to solve problems
ANSWERS
R 13: When the boat comes in
ANSWERS
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HO 1: Space and shape reflection sheet
Use this sheet to record your own reflections and development needs.
Introductory activity
(space and shape in
private, academic,
vocational life)
Carousel of
activities
(assessment /
consolidation /
learning)
Geometric figures
and their properties
Length and area
Construction activity
Area and volume of
3D shapes
Pythagoras and
simple trigonometry
Consolidation
questions
Using concepts of
shape and space to
solve problems
Developmental
tasks
Plenary and close
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HO 2: Trigonometry
A very short history!
Trigonometry, primarily the relationship between the lengths of the sides and the angles in
triangles, dates from the 2nd century BC. Both the Egyptians and Indian mathematicians had
made use of their study of triangles; the former in the context of pyramids* and the later in
the context of holy altars.
(*The seked or slope of a pyramid forms the basis of the developmental task for this section
of the course.)
It was the Greeks though who fully developed the idea and it was Hipparchus who first
compiled tables of trigonometric functions. His key interest was in calculating and predicting
the position of planets and he used imaginary triangles in the sky to do this.
Indian and Arab mathematicians continued the study and it was Hindu mathematicians who
first used sines as we recognise them today.
Modern trigonometry dates from the mid-1500s when its principles were applied to clocks,
engineering, construction, artillery and navigation.
Trigonometry in right angled triangles
Trig is used to work out the lengths and angles in right angled triangles. If you are ‘given’, or
can measure, an angle and a side it is possible to calculate the missing sides. This is still
very important to engineers, surveyors, architects and astronomers today.
The hypotenuse is always the longest side and
is the side opposite the right angle.
O
pp
osi
te
The adjacent side is the side next to the angle
being considered.
Hypotenuse
The opposite side is the side opposite the
angle being considered.
θ
Adjacent
(NB the adjacent and opposite sides are
named with respect to a given angle. If the
other angle in the triangle in the diagram had
been marked then the adjacent and opposite
sides would be reversed.)
Trig Ratios
In right angled triangles the ratio of the lengths of sides relate to the angles within the
triangle. So for the above diagram:
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Worked example
In the diagram - find the length of AC.
A
AC is the hypotenuse and we are given the
adjacent side to 30° so we select the trig ratio:
Substitute:
Rearrange:
30°
Carry out calculation:
40cm
B
C
cm
Rearrangement of formula
The following method can be used for rearranging the three trig ratios.
opp
sin
adj
hyp
cos
opp
hyp
tan
adj
From the diagrams we get:
opp = sin x hyp
adj = cos x hyp
opp = tan x adj
sin = opp/hyp
cos = adj/hyp
tan = opp/adj
hyp = opp/sin
hyp = adj/cos
adj = opp/tan
NB The same method can be used for any similarly structured formula, for example
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R 1: Pentominoes
What is a pentomino?

A collection of five equal squares so that at least one side of
each square is coincident with that of an adjacent square.

Rotations, reflections and translations are not permitted.

How many different arrangements can you find?

Compare with a partner and challenge any arrangements you disagree with.

Make notes of any points you want to raise about terminology, area /
perimeter of the pentominoes you discover, challenges you faced.

Look at the solution sheet and try some of the other challenges
This resource is reproduced with the kind permission of the Centre for Innovation in
Mathematics Teaching (CIMT) based at Plymouth University www.cimt.org.uk
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R 2: Classifying shapes
From the Standards Unit materials – session SS1:
http://tlp.excellencegateway.org.uk/pdf/mat_imp_02.pdf See TN 2 for instructions.
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R 3: Dissecting a square
From the Standards Unit materials – session SS3:
http://tlp.excellencegateway.org.uk/pdf/mat_imp_02.pdf
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R 4: Transformations
From the Standards Unit materials (session SS7):
http://tlp.excellencegateway.org.uk/teachingandlearning/downloads/default.aspx#/math
On the sheet below, draw the image of the L-shape after the following
transformations:
 Shape A: reflection in x axis
 Shape B: reflection in y axis
 Shape C: reflection in the line y = x
 Shape D: reflection in the line y = -x
 Shape E: translation of -4 units vertically
 Shape F: rotation of 90º clockwise about the origin
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Extension

Make up your own, more complex, transformations, record the image and
label it with the transformation (for example, use a vector translation (xy), use
shapes which straddle the line of reflection, use a point of rotation other than
the origin).

Use the blank grid below to explore with more complex shapes.
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Transformation cards: Pictures
Link up pairs of Picture cards using the Word cards. Try to end up with a connected
network, using as many cards as possible.
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Transformation cards: Words
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R 5: Geometric figures and concepts
See TN 3 for instructions.
Area
Perimeter
2D shapes
(excluding
quadrilaterals)
Regular and
irregular shapes
Internal and
external angles
Transformations
3D shapes
Tessellations
Ratios of length,
area and volume
Similar shapes
Congruent
shapes
Triangles
Circles
Angles
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Quadrilaterals
(names and
properties)
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R 6: Evaluating statements about area and perimeter
From the Standards Unit materials – session SS4:
http://tlp.excellencegateway.org.uk/pdf/mat_imp_02.pdf
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R 7: Hints for solutions about area and perimeter statements
From the Standards Unit materials – session SS4:
http://tlp.excellencegateway.org.uk/pdf/mat_imp_02.pdf
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R 8: Pythagoras’ theorem – consolidation question
1. A ladder, 5m long, is resting against a wall as shown.
The base of the ladder is 1.5m from the wall.
How far up the wall does the ladder reach?
5m
1.5 m
2. A plane is coming in to land. It is flying at 4000ft and is currently 2 miles south of
the runway.
What is the actual distance the plane has left to fly before it lands? (1 mile =
5280 ft)
3. A square has a diagonal 14.14cm long.
What is the area of the square?
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R 9: Trigonometry calculations – practice questions
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R 10: The sine rule
For any triangle ABC with sides a, b, c
A
opposite the angles A, B, C
c
B
a
b
c


sin A sin B sin C
b
a
C
The area of such a triangle can be found using the formula: A = ½ ab sinC
Draw a diagram to represent each triangle and then find the missing sides and angles
(where possible).
1.
A = 10º
B = 84º
a = 6cm
2.
C = 125º
B = 17º
b = 2.4cm
3.
B = 62º
b = 8cm
a = 4cm
Much surveying work was dependent on trigonometry (and still is despite advances in
technology) and, without GPS, sea and air navigation would be dependent upon these
principles.
Field sale
A field is coming up for auction and the surveyor needs to describe it accurately for potential
purchasers. The surveyor draws a sketch of the field, which is a quadrilateral, and labels it
ABCD.
Starting at B he measures angle ABD at 21º and angle DBC at 54º. He walks from B to D
and records the distance as 150m. At D he measures the angle ADB as 28º and BDC as
49º.
What will he record in the auction brochure as the lengths of each side of the field and the
area?
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R 11: Using geometry to solve problems
1.2m
30cm
90cm
70cm
30cm
1.7m
The diagram shows the end section of a wooden hen house which is 2m long. If the
recommended space for keeping chicken is 0.18m3, what is the maximum number of
chicken which can be kept in the hen house?
**********************
A company making prefabricated concrete items has an order for 100 concrete posts. Each
is in the shape of a hemisphere surmounted on a cylinder each of radius 15cm. If the total
height of the post is 1.5m, what volume of concrete is required for each post?
The apprentice has ordered 10m3 of concrete to do the job. Will it be enough? What factors
might need to be considered?
**********************
The Olympic committee designed the running track for the 2012 Olympics. They specified
that it must have two parallel straight sides of 100m each and semi-circular ends;
competitors in the 1500m will run four complete circuits.
What area will be available on the inside of the track for field events?
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R 12: Using trigonometry to solve problems
This question is taken from a QCA Key Skills Application of Number test paper and is
reproduced here with kind permission of QCA.
Ice rink
A company is contracted to build a temporary ice rink for a city council for a special event.
Company workers will construct a level platform for the proposed ice rink. The length of the
platform will be 36 metres. It will be built over ground that slopes at an angle of 4°. To make
the platform level, it will stand on vertical supports.
The simplified diagram below shows a cross-section of the platform and one of its supports.
Cliff height
Simon is looking for cliffs from which to do some paragliding. He requires a height of around
145m for a good flight. He is estimating the height of some cliffs from the beach below and
takes the following readings.
From his first position, which he estimates to be 200m from the foot of the cliff, the angle of
elevation of the top of the cliff is 35º.
He takes 50 steps towards the cliff and the angle of elevation is now 45º.
Would you advise him to fly here?
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R 13: When the boat comes in
This problem is reproduced with kind permission of the ‘1000 problems to enjoy’ website:
http://1000problems.org/
The problem
We all know there’s something unexpected about what happens when you pull
something towards you that you can’t quite see ... well, this is especially true for
boats.
Imagine you are on a jetty, and you are pulling in a boat that is floating on the water
some way away. The rope comes up over the edge of the jetty, and lies along the
jetty as you pull the boat in.
As you pull in 10 metres of rope, the boat clearly moves in too, but does it move
exactly 10 metres, more than 10 metres or less than 10 metres?
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R 1: Pentominoes
ANSWERS
There are 12 pentominoes:
They can be used in further spatial challenges:
The enclosure problem
What area can you enclose using all 12?
Space Filling:
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Enlargement:
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R 3: Dissecting a square
Piece
Fraction
Percentage
ANSWERS
Reason
A
1/4
25%
B
1/16
6.25%
e.g. B is ¼ of a ¼, which is 1/16
C
3/16
18.75%
e.g. C = A – B. ¼= 4/16 – 1/16 = 3/16
D
5/16
31.25%
e.g. D is ¼, plus ¼ of another ¼ - i.e. 1/16
4/16 + 1/16 = 5/16
E
1/16
6.25%
e.g. D = 5/16 and F = 1/8 = 2/16, so D + F =
7/16, so E must be 1/16 to make up ½
F
1/8
12.5%
e.g. F is ½ of ¼, which is 1/8
(NB Participants may give other valid reasons as well as those shown.)
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R 4: Transformations
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ANSWERS
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R 6 / R 7: Evaluating statements about area and perimeter
ANSWERS
The following statements are always true:

D, E, F, G, H
The following statements are sometimes true:
 A, B
The following statements are never true:
 C
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R 8: Pythagoras’ theorem – consolidation questions ANSWERS
1. 4.77m
(52 = 1.52 + h2 so h2 = 52 - 1.52)
2. 11 292 feet (d2 = 40002 + 105602)
3. 99.97cm2 Let side of square = s. Then 14.142 = s2 + s2 so s2 = ½ (14.142)
In this case s2 is also equal to the area because the shape is a square.
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R 9: Trigonometry calculations
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ANSWERS
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R 10: The sine rule
ANSWERS
1.
C = 86o
b = 34.36cm
c = 34.47cm
2.
A = 38o
a = 5.05cm
c = 6.72cm
3.
A = 26o
C = 92o
c = 9.06cm
Field sale
AB = 93.3m
BC = 116.2m
CD = 124.5m
DA = 71.2m
Area
= 2506.75 + 7048.06
= 9554.81 or 9555m2
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R 11: Using geometry to solve problems
ANSWERS
Hen house
Volume of hen house is the area of the end section multiplied by the length.
End section can be divided into a triangle (0.12 m2), 2 rectangles (0.36 m2 and 0.51 m2) and
a trapezium (0.145 m2).
(The formula for the area of a trapezium is the average of the parallel sides multiplied by the
perpendicular distance between them.)
Total area of end section = 1.135m2
Volume = 1.135 x 2 = 2.27m3
Max number of hens = 2.27 ÷ 0.18 = 12.6
Max number = 12 hens
Concrete posts
Draw a diagram of one post. Height of cylinder = 1.5m – 0.15m = 1.35m
Vol of hemisphere = ½ (4/3 π r3 ) = 2/3 π r3 = 7.068 x 10-3 m3
Vol of cylinder = π r2 h = 0.0954 m3
Total volume of one post = 0.11 m3 (to 2 dps)
Apprentice has not ordered enough as will need at least 11 m3. Perhaps the apprentice did a
rough calculation which resulted in an under-estimate. Also need to consider waste and
possible breakages. (Construction industry normally adds 5% wastage.)
Running track
Draw a diagram.
Need to calculate distance between parallel sides which is the diameter of the semi-circular
ends.
Total length of one circuit = 1500 ÷ 4 = 375m
Length of semi-circular ends = 175m
Circumference of a circle = 2πr of πD (r = radius, D = diameter)
So πD = 175 and D = 175 ÷ π = 55.7m so r = 27.85m
Area for field events = π r2 + 100D
= 2437 + 5570
= 8007 m2
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R 12: Using trigonometry to solve problems
ANSWERS
Ice rink
tan 4 = o = h
a
36
h = 36 x tan 4
h = 2.52m
Cliff height
Cliff (h)
45
35
50 steps
200m
tan 35 = o = h
a
200
h = 200 x tan 35
Would you advise him to fly?
h = 140 m
All measurements are based on estimates which
may be quite a long way off reality. It could be a
matter of life and death so I would suggest he
didn’t fly. It would be better to contact the local
paragliding club and ask them!
Let 1 step = 1metre
Then:
tan 45 = o = h
a
150
h = 150 x tan 45
h = 150 m
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R 13: When the boat comes in
ANSWERS
You can solve this by making up a height for the jetty and using a Pythagoras-type
sledge hammer, but it is very much simpler than that.
As the rope is pulled a distance x, the boat moves from, say, B to C.
So CA = BA - x
For any triangle, the sum of any two sides is greater than the third, so
BC + CA > BA
Hence, BC + (BA - x) > BA
Hence, BC - x > 0, or, equivalently, BC > x
So the boat comes in further than you pull the rope.
Nice, isn’t it?
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